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pattern synonyms

Last edited by Sebastian Graf
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Pattern Synonyms

Most language entities in Haskell can be named so that they can be abbreviated instead of written out in full. This proposal provides the same power for patterns. See the implementation page for implementation details.

See the PatternSynonyms label.

Motivating example

Here is a simple representation of types

data Type = App String [Type]

Using this representations the arrow type looks like App "->" [t1, t2]. Here are functions that collect all argument types of nested arrows and recognize the Int type:

collectArgs :: Type -> [Type]
collectArgs (App "->" [t1, t2]) = t1 : collectArgs t2
collectArgs _ = []

isInt (App "Int" []) = True
isInt _ = False

Matching on App directly is both hard to read and error prone to write.

The proposal is to introduce a way to give patterns names:

pattern Arrow t1 t2 = App "->" [t1, t2]
pattern Int = App "Int" []

And now we can write

collectArgs :: Type -> [Type]
collectArgs (Arrow t1 t2) = t1 : collectArgs t2
collectArgs _ = []

isInt Int = True
isInt _ = False

Here is a second example from pigworker on Reddit. Your basic sums-of-products functors can be built from this kit.

newtype K a        x  = K a
newtype I          x  = I x
newtype (:+:) f g  x  = Sum (Either (f x) (g x))
newtype (:*:) f g  x  = Prod (f x, g x)

and then you can make recursive datatypes via

newtype Fix f = In (f (Fix f))

e.g.,

type Tree = Fix (K () :+: (I :*: I))

and you can get useful generic operations cheaply because the functors in the kit are all Traversable, admit a partial zip operation, etc.

You can define friendly constructors for use in expressions

leaf :: Tree
leaf = In (Sum (Left (K ())))
node :: Tree -> Tree -> Tree
node l r = In (Sum (Right (Prod (I l, I r))))

but any Tree-specific pattern matching code you write will be wide and obscure. Turning these definitions into pattern synonyms means you can have both readable type-specific programs and handy generics without marshalling your data between views.

Uni-directional (pattern-only) synonyms

The simplest form of pattern synonyms is the one from the examples above. The grammar rule is:

pattern conid varid1 ... varidn <- pat

pattern varid1 consym varid2 <- pat

  • Each of the variables on the left hand side must occur exactly once on the right hand side
  • Pattern synonyms are not allowed to be recursive. Cf. type synonyms.
There have been several proposals for the syntax of defining pattern-only synonyms:
  • pattern conid varid1 ... varidn ~ pat
  • pattern conid varid1 ... varidn := pat
  • pattern conid varid1 ... varidn -> pat
  • pattern conid varid1 ... varidn <- pat

Pattern synonyms can be exported and imported by prefixing the conid with the keyword pattern:

module Foo (pattern Arrow) where ...

This is required because pattern synonyms are in the namespace of constructors, so it's perfectly valid to have

data P = C
pattern P = 42

You may also give a type signature for a pattern, but as with most other type signatures in Haskell it is optional:

pattern conid :: type

E.g.

pattern Arrow :: Type -> Type -> Type
pattern Arrow t1 t2 <- App "->" [t1, t2]

Together with ViewPatterns we can now create patterns that look like regular patterns to match on existing (perhaps abstract) types in new ways:

import qualified Data.Sequence as Seq

pattern Empty <- (Seq.viewl -> Seq.EmptyL)
pattern x :< xs <- (Seq.viewl -> x Seq.:< xs)
pattern xs :> x <- (Seq.viewr -> xs Seq.:> x)

Simply-bidirectional pattern synonyms

In cases where pat is in the intersection of the grammars for patterns and expressions (i.e. is valid both as an expression and a pattern), the pattern synonym can be made bidirectional, and can be used in expression contexts as well. Bidirectional pattern synonyms have the following syntax:

pattern conid varid1 ... varidn = pat

pattern varid1 consym varid2 = pat

For example, the following two pattern synonym definitions are rejected, because they are not bidirectional (but they would be valid as pattern-only synonyms)

pattern ThirdElem x = _:_:x:_
pattern Snd y = (x, y)

since the right-hand side is not a closed expression of {x} and {y} respectively.

In contrast, the pattern synonyms for Arrow and Int above are bidirectional, so you can e.g. write:

arrows :: [Type] -> Type -> Type
arrows = flip $ foldr Arrow

Explicitly-bidirectional pattern synonyms

What if you want to use Succ in an expression:

    pattern Succ n <- n1 | let n = n1 -1, n >= 0

It's clearly impossible since its expansion is a pattern that has no meaning as an expression. Nevertheless, if we want to make what looks like a constructor for a type we will often want to use it in both patterns and expressions. This is the rationale for the most complicated synonyms, the bidirectional ones. They provide two expansions, one for patterns and one for expressions.

pattern conid varid1 ... varidn <- pat where cfunlhs rhs

where cfunlhs is like funlhs, except that the functions symbol is a conid instead of a varid.

Example, using ViewPatterns:

pattern Succ n <- ((\x -> (x-1) <$ guard (x > 0)) -> Just n)
 where
  Succ n = n + 1

The first part as is before and describes the expansion of the synonym in patterns. The second part describes the expansion in expressions.

fac (Succ n) = Succ n * fac n 
fac 0 = 1

Associated pattern synonyms

Just like data types and type synonyms can be part of a class declaration, it would be possible to have pattern synonyms as well.

Example:

class ListLike l where
  pattern Nil :: l a
  pattern Cons :: a -> l a -> l a
  isNil :: l a -> Bool
  isNil Nil = True
  isNil (Cons _ _) = False
  append :: l a -> l a -> l a

instance ListLike [] where
  pattern Nil = []
  pattern Cons x xs = x:xs
  append = (++)

headOf :: (ListLike l) => l a -> Maybe a
headOf Nil = Nothing
headOf (Cons x _) = Just x

One could go one step further and leave out the pattern keyword to obtain associated constructors, which are required to be bidirectional. The capitalized identifier would indicate that a pattern synonym is being defined. For complicated cases one could resort to the where syntax (shown above).

TODO: Syntax for associated pattern synonym declarations to discern between pattern-only and bidirectional pattern synonyms

Static semantics

A unidirectional pattern synonym declaration has the form

pattern P var1 var2 ... varN <- pat

The formal pattern synonym arguments var1, var2, ..., varN are brought into scope by the pattern pat on the right-hand side. The declaration brings the name P as a pattern synonym into the module-level scope.

The pattern synonym P is assigned a pattern type of the form

pattern P :: CReq => CProv => t1 -> t2 -> ... -> tN -> t

where t1, ..., tN are the types of the parameters var1, ..., varN, t is the simple type (with no context) of the thing getting matched, and CReq and CProv are type contexts.

CProv can be omitted if it is empty. If CReq is empty, but CProv is not, () is used. The following example shows cases:

data Showable where
    MkShowable :: (Show a) => a -> Showable

-- Required context is empty, but provided context is not
pattern Sh :: () => (Show a) => a -> Showable
pattern Sh x <- MkShowable x

-- Provided context is empty
pattern One :: (Num a, Eq a) => a
pattern One <- 1

A pattern synonym can be used in a pattern if the instatiated (monomorphic) type satisfies the constraints of CReq. In this case, it extends the context available in the right-hand side of the match with CProv, just like how an existentially-typed data constructor can extend the context.

As with function and variable types, the pattern type signature can be inferred, or it can be explicitly written out on the program.

Here's a more complex example. Let's look at the following definition:

{-# LANGUAGE PatternSynonyms, GADTs, ViewPatterns #-}
module ShouldCompile where

data T a where
	MkT :: (Eq b) => a -> b -> T a

f :: (Show a) => a -> Bool

pattern P x <- MkT (f -> True) x

Here, the inferred type of P is

pattern P :: (Show a) => (Eq b) => b -> T a

A bidirectional pattern synonym declaration has the form

pattern P var1 var2 ... varN = pat

where both of the following are well-typed declarations:

pattern P1 var1 var2 ... varN <- pat

P2 = \var1 var2 ... varN -> pat

In this case, the pattern type of P is simply the pattern type of P1, and its expression type is the type of P2. The name P is brought into the module-level scope both as a pattern synonym and as an expression.

Dynamic semantics

A pattern synonym occurance in a pattern is evaluated by first matching against the pattern synonym itself, and then on the argument patterns. For example, given the following definitions:

pattern P x y <- [x, y]

f (P True True) = True
f _             = False

g [True, True] = True
g _            = False

the behaviour of f is the same as

f [x, y] | True <- x, True <- y = True
f _                             = False

Because of this, the eagerness of f and g differ:

*Main> f (False:undefined)
*** Exception: Prelude.undefined
*Main> g (False:undefined)
False

This is because we generate the matching function at the definition site.

Typed pattern synonyms

So far patterns only had syntactic meaning. In comparison Ωmega has typed pattern synonyms, so they become first class values. For bidirectional pattern synonyms this seems to be the case

data Nat = Z | S Nat deriving Show
pattern Ess p = S p

And it works:

*Main> map S [Z, Z, S Z]
[S Z,S Z,S (S Z)]
*Main> map Ess [Z, Z, S Z]
[S Z,S Z,S (S Z)]

Branching pattern-only synonyms

*N.B. this is a speculative suggestion! *

Sometimes you want to match against several summands of an ADT simultaneously. E.g. in a data type of potentially unbounded natural numbers:

data Nat = Zero | Succ Nat
type UNat = Maybe Nat -- Nothing meaning unbounded

Conceptually Nothing means infinite, so it makes sense to interpret it as a successor of something. We wish it to have a predecessor just like Just (Succ Zero)!

I suggest branching pattern synonyms for this purpose:

pattern S pred <- pred@Nothing | pred@(Just a <- Just (Succ a))
pattern Z = Just Zero

Here pred@(Just a <- Just (Succ a)) means that the pattern invocation S pred matches against Just (Succ a) and - if successful - binds Just a to pred.

This means we can syntactically address unbound naturals just like bounded ones:

greetTimes :: UNat -> String -> IO ()
greetTimes Z _ = return ()
greetTimes (S rest) message = putStrLn message >> greetTimes rest message

As a nice collateral win this proposal handles pattern Name name <- Person name workplace | Dog name vet too.

Record Pattern Synonyms

See PatternSynonyms/RecordPatternSynonyms

Associating synonyms with types

See PatternSynonyms/AssociatingSynonyms

COMPLETE pragmas

See PatternSynonyms/CompleteSigs